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Q.
If for the non-singular matrix $A, A^{2} = I$, then find $A^{-1}$.
Determinants
Solution:
Given, $A^{2} = I $
Since, A is non-singular matrix
$\therefore \quad\left|A\right|\ne0, so, A^{-1}$ exists.
Multiplying by $A^{-1}$ on both sides, we get
$A^{-1}\left(A^{2}\right)=A^{-1}\left(I\right)$
$\Rightarrow \quad\left(A^{-1}A\right)A=A^{-1} \Rightarrow \, IA=A^{-1} \quad\left(\because\quad A^{-1}A=I\right)$
$\therefore \quad A^{-1}=A \quad\left(\because\quad IA=A\right)$